Skip to main content

Full text of "Treatise On Analysis Vol-Ii"

See other formats


(5)          (fJL ~ A)"

(" Christoffel-Darboux formula");

(c)   Denote by Kn(A), where A e C and </A > 0, the set of complex numbers w
such that

Show that Kn(A) is a closed disk contained in the half-plane Sw > 0, with center at
the point

Pn(A)Pn-1(A) - P.-
and radius

the frontier of Kn(A) being the closure of the set of points


where reR. We have Kn+i(A) c KW(A), and the frontiers of these two disks have a
point in common. The intersection Koo(A) of these disks consists of a single point if

and otherwise is a disk of radius

(d)    Show that the following three properties are equivalent:

(a)   The operator H deduced from the Jacobi matrix / is self-adjoint,
(j8)   There exists only one positive measure v which extends the linear form aF
(in other words, the "moment problem" for the sequence (cn) has only
one solution).
(y)   The set Koo(A) consists of a single point, for some A 6 C with J*A > 0 (and

consequently for all A e C with J\ > 0).

(To show that (^8) and (y) are equivalent, consider the matrices / with n rows and
n columns obtained by deleting the rows and columns with indices >n in J\ consider
the corresponding problem in C", and then pass to the limit.)

(e)    If the equivalent conditions in (d) are satisfied, then the Pn form a total system                                   fjizn = bn. lzn.i-}-anzn^rbnzn+l,